Question

Find vector fields$\mathbf{A}\mathbf{}$ such that $\mathbf{V}\mathbf{}\mathbf{=}\mathbf{}\mathbf{curl}\mathbf{}\mathbf{A}$for each given V.$\mathbf{V}=(\mathbf{y}+\mathbf{z})\mathbf{i}+(\mathbf{x}-\mathbf{z})\mathbf{j}+\left({\mathbf{x}}^2+{\mathbf{y}}^2\right)\mathbf{k}$

Short answer

The vector field derived is$\mathrm{A}=\mathrm{i}\left(\mathrm{xz}-\frac{{\mathrm{y}}^3}{3}\right)+\mathrm{j}\left(-\mathrm{yz}+\frac{{\mathrm{x}}^3}{3}\right)+\mathrm{k}(\mathrm{x}+\mathrm{y})\mathrm{z}+\mathrm{\nabla }\mathrm{u}$.

Step-by-step solution

Given Information.

The given equation is$\mathrm{V}=(\mathrm{y}+\mathrm{z})\mathrm{i}+(\mathrm{x}-\mathrm{z})\mathrm{j}+\left({\mathrm{x}}^2+{\mathrm{y}}^2\right)\mathrm{k}$.

Definition of vector.

A quantity that has magnitude as well as direction is called a vector. It is typically denoted by an arrow in which the head determines the direction of the vector and the length determines its magnitude.

Find the vector field.

In this problem find vector $\mathrm{A}$, whose curl gives the vector

$\mathrm{V}=(\mathrm{y}+\mathrm{z})\mathrm{i}+(\mathrm{x}-\mathrm{z})\mathrm{j}+\left({\mathrm{x}}^2+{\mathrm{y}}^2\right)\mathrm{k}$

The components of vector are given in terms of the derivatives of $\mathrm{A}$.

$\frac{\mathrm{\partial }{\mathrm{A}}_{\mathrm{z}}}{\mathrm{\partial }\mathrm{y}}-\frac{\mathrm{\partial }{\mathrm{A}}_{\mathrm{y}}}{\mathrm{\partial }\mathrm{z}}=\mathrm{y}+\mathrm{z}$

$\frac{\mathrm{\partial }{\mathrm{A}}_x}{\mathrm{\partial }\mathrm{y}}-\frac{\mathrm{\partial }{\mathrm{A}}_z}{\mathrm{\partial }\mathrm{z}}=x-\mathrm{z}$

$\frac{\mathrm{\partial }{\mathrm{A}}_{\mathrm{y}}}{\mathrm{\partial }\mathrm{x}}-\frac{\mathrm{\partial }{\mathrm{A}}_{\mathrm{x}}}{\mathrm{\partial }\mathrm{y}}={\mathrm{x}}^2+{\mathrm{y}}^2$

Observe the first two equation ${\mathrm{A}}_{\mathrm{z}}=\mathrm{Z}(\mathrm{x}+\mathrm{y})$, and choose the equation.

The first equation then gives the equation $-\frac{\mathrm{\partial }{\mathrm{A}}_{\mathrm{y}}}{\mathrm{\partial }\mathrm{z}}=\mathrm{y}$.

Integrate to obtain the equation ${\mathrm{A}}_{\mathrm{y}}=-\mathrm{yz}+\mathrm{f}(\mathrm{x},\mathrm{y})$.